Joshua Moerman
11 years ago
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3c8b2a1fee
7 changed files with 234 additions and 0 deletions
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.DS_Store |
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*.pdf |
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build |
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\section{Definitions} |
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\label{sec:definitions} |
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\subsection{Graded algebra} |
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In this section $\k$ will be any commutative ring. We will recap some of the basic definitions of commutative algebra in a graded setting. By \emph{linear}, \emph{module}, \emph{tensor product}, \dots we always mean $\k$-linear, $\k$-module, tensor product over $\k$, \dots. |
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\begin{definition} |
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A \emph{graded module} $M$ is a family of modules $\{M_n\}_{n\in\Z}$. An element $x \in M_n$ is called a \emph{homogenous element} and said to be of \emph{degree $\deg{x} = n$}. We will often identify $M = \bigoplus_{n \in \Z} M_n$. |
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\end{definition} |
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For an arbitrary module $M$ we can consider the graded module $M[0]$ \emph{concentrated in degree $0$} defined by setting $M[0]_0 = M$ and $M[0]_n = 0$ for $i \neq 0$. If clear from the context we will denote this graded module by $M$. In particular $\k$ is a graded module concentrated in degree $0$. |
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\begin{definition} |
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A linear map $f: M \to N$ between graded modules is \emph{graded of degree $p$} if it respects the grading, i.e. $\restr{f}{M_n} : M_n \to N_{n+p}$. |
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\end{definition} |
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\begin{definition} |
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The graded maps $f: M \to N$ between graded modules can be arranged in a graded module by defining: |
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$$ \Hom{gr}{M}{N}_n = \{ f: M \to N \I f \text{ is graded of degree } n \}. $$ |
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\end{definition} |
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Note that not all linear maps can be decomposed into a sum of graded maps. In other words $\Hom{gr}{M}{N} \subset \Hom{}{M}{N}$ might not be equal. |
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Recall that the tensor product of modules distributes over direct sums. So if $M = \bigoplus_{n \in \Z} M_n$ and $N = \bigoplus_{n \in \Z} N_n$, then |
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$$ M \tensor N \iso \bigoplus_{n \in Z} \bigoplus_{m \in Z} M_m \tensor N_n \iso \bigoplus_{n \in Z} \bigoplus_{i + j = n} M_i \tensor N_j. $$ |
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This defines a natural grading on the tensor product. |
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\begin{definition} |
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The graded tensor product is defined as: |
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$$ (M \tensor N)_n = \bigoplus_{i + j = n} M_i \tensor N_j. $$ |
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\end{definition} |
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The graded modules together with graded maps of degree $0$ form the category $\grMod{\k}$ of graded modules. Together with the tensor product and the ground ring, $(\grMod{\k}, \tensor, \k)$ is a monoidal category. This now dictates the definition of a graded algebra. |
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\begin{definition} |
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A \emph{graded algebra} consists of a graded module $A$ together with two graded maps of degree $0$: |
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$$ \mu: A \tensor A \to A \quad\text{ and }\quad \eta: k \to A $$ |
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such that $\mu$ is associative and $\eta$ is a unit for $\mu$. |
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A graded map between two graded algebra will be called \emph{graded algebra map} if the map is compatible with the multiplication and unit. |
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\end{definition} |
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Again these objects form a category, denoted as $\grAlg{\k}$. |
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\begin{definition} |
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A graded algebra $A$ is \emph{commutative} if for all $x, y \in A$ |
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$$ xy = (-1)^{\deg{x}\deg{y}}yx. $$ |
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\end{definition} |
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\subsection{Differential graded algebra} |
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Now define differentials... and the categories $\cat{DGA}_\k, \cat{CGDA}_\k$. |
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Note that a monoidal object of differential graded modules is the same as a graded algebra with a differential. |
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Conclude with (co)chain complexes and (co)chain (co)algebras. |
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\subsection{Model categories} |
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.PHONY: thesis fast images |
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# We don want to pollute the root dir, so we use a build dir
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# http://tex.stackexchange.com/questions/12686/how-do-i-run-bibtex-after-using-the-output-directory-flag-with-pdflatex-when-f
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thesis: |
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mkdir -p build |
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cp references.bib build/ |
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pdflatex -output-directory=build thesis.tex |
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cd build; bibtex thesis |
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pdflatex -output-directory=build thesis.tex |
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pdflatex -output-directory=build thesis.tex |
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cp build/thesis.pdf ./ |
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fast: |
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mkdir -p build |
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pdflatex -output-directory=build thesis.tex |
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cp build/thesis.pdf ./ |
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images: |
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mkdir -p build |
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pdflatex -output-directory=build images.tex |
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pdflatex -output-directory=build images.tex |
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scp build/images.pdf moerman@stitch.science.ru.nl:~/wvlt_images.pdf |
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ssh moerman@stitch.science.ru.nl 'pdf2svg wvlt_images.pdf wvlt_images.svg' |
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scp moerman@stitch.science.ru.nl:~/wvlt_images.svg ./images.svg |
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% clickable tocs |
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\usepackage{hyperref} |
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% floating figures |
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\usepackage{float} |
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\usepackage{listings} |
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\usepackage{tikz} |
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\usetikzlibrary{matrix, arrows, decorations} |
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\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto} |
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\usepackage{graphicx} |
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\graphicspath{ {./images/} } |
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\usepackage{caption} |
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\usepackage{subcaption} |
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% Matrices have a upper bound for its size |
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\setcounter{MaxMatrixCols}{20} |
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% Remove trailing `contents` after toc |
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\renewcommand{\contentsname}{} |
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% for the fib arrow |
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\usepackage{amssymb} |
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% mathbb for lowercase |
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\usepackage{bbm} |
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% for slanted text/symbols |
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\usepackage{slantsc} |
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\DeclareMathOperator*{\colim}{colim} |
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\DeclareMathOperator*{\tensor}{\otimes} |
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\DeclareMathOperator*{\bigtensor}{\bigotimes} |
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\newcommand{\N}{\mathbb{N}} |
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\newcommand{\Np}{{\mathbb{N}^{>0}}} |
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\newcommand{\Z}{\mathbb{Z}} |
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\newcommand{\R}{\mathbb{R}} |
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\renewcommand{\k}{\mathbbm{k}} |
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\newcommand{\cat}[1]{\mathbf{#1}} |
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\newcommand{\Set}{\cat{Set}} |
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\newcommand{\sSet}{\cat{sSet}} |
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\newcommand{\Top}{\cat{Top}} |
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\newcommand{\DELTA}{\cat{\Delta}} |
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\newcommand{\grMod}[1]{\cat{gr-{#1}Mod}} |
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\newcommand{\grAlg}[1]{\cat{gr-{#1}Alg}} |
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\newcommand{\Hom}[3]{\mathbf{Hom}_{#1}(#2, #3)} |
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\newcommand{\id}{\mathbf{id}} |
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\newcommand{\I}{\,\mid\,} |
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\newcommand{\del}{\partial} % boundary |
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\newcommand{\iso}{\cong} % isomorphic |
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\newcommand{\eq}{\sim} % homotopic |
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\newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}} % arrow with name |
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\newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}} % mapsto with name |
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\newcommand{\cof}{\hookrightarrow} % cofibration |
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\newcommand{\fib}{\twoheadrightarrow} % fibration |
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\newcommand{\we}{\tot{\simeq}} % weak equivalence |
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\renewcommand{\deg}[1]{|{#1}|} |
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\newcommand\restr[2]{{% we make the whole thing an ordinary symbol |
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\left.\kern-\nulldelimiterspace % automatically resize the bar with \right |
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#1 % the function |
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\vphantom{\big|} % pretend it's a little taller at normal size |
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\right|_{#2} % this is the delimiter |
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}} |
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\newcommand{\todo}[1]{ |
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\addcontentsline{tdo}{todo}{\protect{#1}} |
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$\ast$ \marginpar{\tiny $\ast$ #1} |
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} |
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\theoremstyle{plain} |
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\newtheorem{theorem}{Theorem}[section] |
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\newtheorem{proposition}[theorem]{Proposition} |
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\newtheorem{lemma}[theorem]{Lemma} |
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\newtheorem{corollary}[theorem]{Corollary} |
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\theoremstyle{definition} |
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\newtheorem{definition}[theorem]{Definition} |
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\newtheorem{example}[theorem]{Example} |
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\newcommand*{\thead}[1]{\multicolumn{1}{c}{\bfseries #1}} |
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@book{felix, |
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title={Rational homotopy theory}, |
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author={F{\'e}lix, Yves and Halperin, Steve and Thomas, Jean-Claude}, |
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volume={205}, |
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year={2001}, |
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publisher={Springer} |
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} |
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@book{bous, |
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title={On PL de Rham theory and rational homotopy type}, |
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author={Bousfield, Aldridge Knight and Gugenheim, Victor KAM}, |
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volume={179}, |
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year={1976}, |
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publisher={American Mathematical Soc.} |
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} |
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@article{hess, |
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title={Rational homotopy theory: a brief introduction}, |
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author={Hess, Kathryn and others}, |
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journal={Contemporary Mathematics}, |
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volume={436}, |
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pages={175}, |
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year={2007}, |
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publisher={Providence, RI; American Mathematical Society; 1999} |
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} |
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% lesser margins |
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\usepackage{geometry} |
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\geometry{a4paper} |
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\geometry{twoside=false} |
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% no indent, but vertical spacing |
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\usepackage[parfill]{parskip} |
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\setlength{\marginparwidth}{2cm} |
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\documentclass[a4paper, 11pt]{amsart} |
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\input{style} |
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\input{preamble} |
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\title{Rational Homotopy Theory} |
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\author{Joshua Moerman} |
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\begin{document} |
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\maketitle |
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\input{Definitions} \newpage |
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\nocite{*} |
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\bibliographystyle{alpha} |
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\bibliography{references} |
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\end{document} |
Reference in new issue